\[\begin{align} S & = a + ar + ar^{2} + ar^{3} + ... + ar^{n-1} \\ & = a\frac{1 - r^{n}}{1 - r} \end{align}\]
\[\begin{align} S & = Ir^{n} + a\frac{1 - r^{n}}{1 - r} \\ n & = \frac{ \ln \Bigg[ \frac{S(1-r)-a}{I(1-r)-a} \Bigg] }{\ln r} \end{align}\]